Dynamical Analysis for High-Order Delayed Hopfield Neural Networks with Impulses

Abstract

The global exponential stability and uniform stability of the equilibrium point for high-order delayed Hopfield neural networks with impulses are studied. By utilizing Lyapunov functional method, the quality of negative definite matrix, and the linear matrix inequality approach, some new stability criteria for such system are derived. The results are related to the size of delays and impulses. Two examples are also given to illustrate the effectiveness of our results.

1. Introduction

In the last several years, Hopfield neural networks (HNNs) have received especially considerable attention due to their extensive applications in solving optimization problem, traveling salesman problem, and many other subjects, see [1–17]. However such neural networks are shown to have limitations such as limited capacity when used in pattern recognition problems, see [2, 3]. This led many researchers to use neural networks with high order connections. The high-order neural networks have stronger approximation property, faster convergence rate, greater storage capacity, and higher fault tolerance than lower-order neural networks. Recently, various results on stability of high-order delayed HNN are obtained, see [11–15]. For example, Lou and Cui [13] studied the global asymptotic stability of high-order HNN with time-varying delays by using Lyapunov method, linear matrix inequality (LMI), and analytic technique as follows: 𝑥^′_𝑖(𝑡)=−𝑐_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡+_{𝑛𝑛𝑗=1𝑙=1})𝑇_𝑖𝑗𝑙𝑔_𝑙𝑥_𝑙𝑡−𝜏_𝑙(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡)+𝐼_𝑖,𝑡≥𝑡₀,𝑖=1,2,…,𝑛. (1.1) But the authors only obtained some global asymptotic stability criteria for the above high-order HNN. Those results cannot ensure the global exponential stability of the equilibrium point. It is well known that global exponential stability plays an important role in many areas such as designs and applications of neural networks and synchronization in secure communication [5, 17–23]. One purpose of this paper is to improve the results in [13]. We obtain several new criteria on global exponential stability and uniform stability for the above high-order HNN.

On the other hand, it is well known that the artificial electronic networks are subject to instantaneous perturbations and experience change of the state abruptly, that is, do exhibit impulsive effects. Such systems are described by impulsive differential systems which have been used successfully in modeling many practical problems arisen in the fields of natural sciences and technology, see [12, 24–30]. Hence, it is very important and, in fact, necessary to investigate the issue of the stability of high-order delayed HNN with impulses. However, to the best of the authors’ knowledge, there are few results on the stability of high-order delayed HNN with impulses. In [12], Liu et al. obtained some sufficient conditions for ensuring global exponential stability of impulsive high order HNN with time-varying delays by using the method of Lyapunov functions.

The purpose of this paper is to present some new criteria concerning the global exponential stability and uniform stability for a class of high-order delayed HNN with impulses by utilizing Lyapunov functional method, the quality of negative definite matrix, and the linear matrix inequality approach. The conditions on impulses are different from that presented in [12]. The effects of impulses and delays on the solutions are stressed here. As a special case, several new criteria on global exponential stability and uniform stability for the corresponding high-order HNN without impulses (see [13]) are obtained. To illustrate the validity of those results, two examples are given to illustrate the effectiveness of the results obtained.

2. Preliminaries

Let ℝ denote the set of real numbers, ℝ₊ the set of nonnegative real numbers, ℤ₊ the set of positive integers, and ℝ^𝑛 the 𝑛-dimensional real space equipped with the Euclidean norm ||⋅||.

Consider the following high-order delayed HNN model with impulses 𝑥^′_𝑖(𝑡)=−𝑐_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑔_𝑗𝑥_𝑗((𝑡+𝑡−𝜏))_{𝑛𝑛𝑗=1𝑙=1}𝑇_𝑖𝑗𝑙𝑔_𝑙𝑥_𝑙(𝑡−𝜏(𝑡))𝑔_𝑗𝑥_𝑗(𝑡−𝜏(𝑡))+𝐼_𝑖,𝑡≠𝑡_𝑘,𝑡≥𝑡₀,Δ𝑥_𝑖|_{𝑡=𝑡𝑘}=𝑥_𝑖𝑡_𝑘−𝑥_𝑖𝑡⁻_𝑘,𝑖∈Λ,𝑘∈ℤ₊, (2.1) where Λ={1,2,…,𝑛}, 𝑛≥2 corresponds to the number of units in a neural network; the impulse times 𝑡_𝑘 satisfy 0≤𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯, lim_𝑘→+∞𝑡_𝑘=+∞; 𝑥_𝑖 corresponds to the membrane potential of the unit 𝑖 at time 𝑡; 𝑐_𝑖 is positive constant; 𝑓_𝑗, 𝑔_𝑗 denote, respectively, the measures of response or activation to its incoming potentials of the unit 𝑗 at time 𝑡 and 𝑡−𝜏(𝑡); 𝑇_𝑖𝑗𝑙 is the second-order synaptic weights of the neural networks; constant 𝑎_𝑖𝑗 denotes the synaptic connection weight of the unit 𝑗 on the unit 𝑖 at time 𝑡; constant 𝑏_𝑖𝑗 denotes the synaptic connection weight of the unit 𝑗 on the unit 𝑖 at time 𝑡−𝜏(𝑡); 𝐼_𝑖 is the input of the unit 𝑖;𝜏(𝑡) is the transmission delay such that 0<𝜏(𝑡)≤𝜏 and 𝜏̇(𝑡)≤𝜌<1, 𝑡≥𝑡₀; 𝜏, 𝜌 are constants.

The initial conditions associated with system (2.1) are of the form 𝑥(𝑠)=𝜙(𝑠),𝑠∈𝑡₀−𝜏,𝑡₀, (2.2) where 𝑥(𝑠)=(𝑥₁(𝑠),𝑥₂(𝑠),…,𝑥_𝑛(𝑠))^𝑇, 𝜙(𝑠)=(𝜙₁(𝑠),𝜙₂(𝑠),…,𝜙_𝑛(𝑠))^𝑇∈𝑃𝐶([−𝜏,0],ℝ^𝑛), 𝑃𝐶([−𝜏,0],ℝ^𝑛)={𝜓∶[−𝜏,0]→ℝ^𝑛 is continuous everywhere except at finite number of points 𝑡_𝑘, at which 𝜓(𝑡⁺_𝑘) and 𝜓(𝑡⁻_𝑘) exist and 𝜓(𝑡⁺_𝑘)=𝜓(𝑡_𝑘)}. For 𝜓∈𝑃𝐶([−𝜏,0],ℝ^𝑛), the norm of 𝜓 is defined by ||𝜓||_𝜏=sup_{−𝜏≤𝜃≤0}|𝜓(𝜃)|. For any𝑡₀≥0, let𝑃𝐶_𝛿(𝑡₀)={𝜓∈𝑃𝐶([−𝜏,0],ℝ^𝑛)∶||𝜓||<𝛿}.

Assume that 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)^𝑇 is an equilibrium point of system (2.1). Impulsive operator is viewed as perturbation of the equilibrium point 𝑥^∗ of such system without impulsive effects. We assume that Δ𝑥_𝑖∣_{𝑡=𝑡𝑘}=𝑥_𝑖𝑡_𝑘−𝑥_𝑖𝑡⁻_𝑘=𝑑_𝑘^(𝑖)𝑥_𝑖𝑡⁻_𝑘−𝑥^∗_𝑖,𝑑_𝑘^(𝑖)∈ℝ,𝑖∈Λ,𝑘∈ℤ₊. (2.3)

Since 𝑥^∗ is an equilibrium point of system (2.1), one can derive from system (2.1)-(2.2) that the transformation 𝑦_𝑖=𝑥_𝑖−𝑥^∗_𝑖, 𝑖∈Λ transforms such system into the following system (for more details, please see papers [12, 13]): 𝑦^′(𝑡)=−𝐶𝑦(𝑡)+𝐴𝐹𝑦(𝑡)+𝐵𝐺𝑦(𝑡−𝜏(𝑡))+Γ^𝑇𝑇^⋆𝐺𝑦(𝑡−𝜏(𝑡)),𝑡≠𝑡_𝑘,𝑡≥𝑡₀,𝑦𝑡_𝑘=𝐷_𝑘𝑦𝑡⁻_𝑘,𝑘∈ℤ₊,𝑦𝑡₀+𝜃=𝜑(𝜃),𝜃∈[−𝜏,0], (2.4) where 𝜑(𝜃)=𝑥𝑡₀+𝜃−𝑥^∗,𝑦(𝑡)=𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡)^𝑇,𝑦(𝑡−𝜏(𝑡))=𝑦₁(𝑡−𝜏(𝑡)),𝑦₂(𝑡−𝜏(𝑡)),…,𝑦_𝑛(𝑡−𝜏(𝑡))^𝑇,𝐹𝑦(𝑡)=𝐹₁𝑦₁(𝑡),𝐹₂𝑦₂(𝑡),…,𝐹_𝑛𝑦_𝑛(𝑡)^𝑇,𝐺𝑦(𝑡−𝜏(𝑡))=𝐺₁𝑦₁(𝑡−𝜏(𝑡)),𝐺₂𝑦₂(𝑡−𝜏(𝑡)),…,𝐺_𝑛𝑦_𝑛(𝑡−𝜏(𝑡))^𝑇,𝐹_𝑗𝑦_𝑗(𝑡)=𝑓_𝑗𝑥^∗_𝑗+𝑦_𝑗(𝑡)−𝑓_𝑗𝑥^∗_𝑗,𝐺_𝑗𝑦_𝑗(𝑡−𝜏(𝑡))=𝑔_𝑗𝑥^∗_𝑗+𝑦_𝑗(𝑡−𝜏(𝑡))−𝑔_𝑗𝑥^∗_𝑗,𝐶=diag𝑐₁,𝑐₂,…,𝑐_𝑛,𝐴=𝑎_{𝑖𝑗𝑛×𝑛},𝐵=𝑏_{𝑖𝑗𝑛×𝑛},𝑇_𝑖=𝑇_{𝑖𝑗𝑙𝑛×𝑛},𝑇^⋆=𝑇₁+𝑇^𝑇₁,𝑇₂+𝑇^𝑇₂,…,𝑇_𝑛+𝑇^𝑇_𝑛^𝑇,Γ=diag[𝜍,𝜍,…,𝜍],𝜍=𝜍₁,𝜍₂,…,𝜍_𝑛^𝑇,𝐷_𝑘=diag1+𝑑_𝑘⁽¹⁾,1+𝑑_𝑘⁽²⁾,…,1+𝑑_𝑘^(𝑛), (2.5) in which 𝜍_𝑙 is a real value between 𝑔_𝑙(𝑥_𝑙(𝑡−𝜏(𝑡))) and 𝑔_𝑙(𝑥^∗_𝑙), 𝑙∈Λ.

Remark 2.1. Obviously, (0,0,…,0)^𝑇 is an equilibrium point of (2.4). Therefore, there exists at least one equilibrium point of system (2.1). So, the stability analysis of the equilibrium point 𝑥^∗ of (2.1) can now be transformed to the stability analysis of the trivial solution 𝑦=0 of (2.4).
In the following, the notations 𝑋^𝑇 and 𝑋⁻¹ mean the transpose of and the inverse of a square matrix 𝑋. We will use the notation 𝑋>0 (or 𝑋<0, 𝑋≥0, 𝑋≤0) to denote that the matrix 𝑋 is a symmetric and positive definite (negative definite, positive semidefinite, negative semidefinite) matrix. Let 𝜆_max(𝑋), 𝜆_min(𝑋), respectively, denote the largest and smallest eigenvalue of matrix 𝑋.
Throughout this paper, we assume that there exist constants 𝜒_𝑖>0,𝑀,𝑁≥0 such that |𝑔_𝑖(𝑥_𝑖)|≤𝜒_𝑖, 𝑖∈Λ, 𝐹^𝑇(𝑦)𝐹(𝑦)≤𝑀𝑦^𝑇𝑦, 𝐺^𝑇(𝑦)𝐺(𝑦)≤𝑁𝑦^𝑇𝑦.

We introduce some definitions as follows.

Definition 2.2 (see [5]). Leting 𝑉∶ℝ₊×ℝ^𝑛→ℝ₊, for any (𝑡,𝑥)∈[𝑡_𝑘−1,𝑡_𝑘)×ℝ^𝑛, the upper right-hand Dini derivative of 𝑉(𝑡,𝑥) along the solution of (2.4) is defined by 𝐷⁺𝑉(𝑡,𝑥)=limsup_ℎ→0+1ℎ𝑉𝑡+ℎ,𝑥+ℎ−𝐶𝑦(𝑡)+𝐴𝐹𝑦(𝑡)+𝐵𝐺𝑦(𝑡−𝜏(𝑡))+Γ^𝑇𝑇^⋆𝐺𝑦(𝑡−𝜏(𝑡))−𝑉(𝑡,𝑥). (2.6)

Definition 2.3 (see [25]). Assume 𝑦(𝑡)=𝑦(𝑡₀,𝜑)(𝑡) is the solution of (2.4) through (𝑡₀,𝜑). Then the zero solution of (2.4) is said to be uniformly stable, if, for any 𝜀>0 and 𝑡₀≥0, there exists some 𝛿=𝛿(𝜀)>0 such that𝜑∈𝑃𝐶_𝛿(𝑡₀) implies ||𝑦(𝑡)||<𝜀, 𝑡≥𝑡₀.

Definition 2.4 (see [5]). The equilibrium point 𝑥^∗ of the system (2.1) is globally exponentially stable, if there exists constant 𝜇>0,𝕄≥1 such that, for any initial value 𝜙, 𝑥𝑡₀,𝜙(𝑡)−𝑥^∗<𝕄𝜙−𝑥^∗_𝜏𝑒^{−𝜇(𝑡−𝑡0)},𝑡≥𝑡₀. (2.7)
Next, in order to obtain our results, we need to establish the following lemma.

Lemma 2.5 (see [13]). For any vectors 𝑎,𝑏∈ℝ^𝑛, the inequality ±2𝑎^𝑇𝑏≤𝑎^𝑇𝑋𝑎+𝑏^𝑇𝑋⁻¹𝑏 (2.8) holds, in which 𝑋 is any 𝑛×𝑛 matrix with 𝑋>0.

Lemma 2.6 (see [31]). Let 𝑋∈𝑅^𝑛×𝑛, then 𝜆_min(𝑋)𝑎^𝑇𝑎≤𝑎^𝑇𝑋𝑎≤𝜆_max(𝑋)𝑎^𝑇𝑎 (2.9) for any 𝑎∈ℝ^𝑛 if 𝑋 is a symmetric matrix.

3. Main Results

In this section, some sufficient delay-dependent conditions of global exponential stability and uniform stability for system (2.1) are obtained.

Theorem 3.1. Assume that there exist constants 𝜀^⋆>0, 𝛿^⋆∈[0,𝜀^⋆) and 𝑛×𝑛 symmetric and positive definite matrices 𝑃, 𝑄₁, 𝑄₂ such that(i)  𝜀^⋆𝑃−𝑃𝐶−𝐶𝑃+𝑃𝐴𝑄₁⁻¹𝐴^𝑇𝑃+𝜆_max𝑄₁𝑀𝐸+𝑁𝜆_max𝑄₂+𝑇^⋆𝑇𝑇^⋆𝐸1−𝜌+𝑒^𝜏𝜀⋆𝑃𝐵𝑄₂⁻¹𝐵^𝑇𝑃+𝑒^𝜏𝜀⋆𝜒²𝑃²≤0, (3.1)where 𝜒=(𝜒₁,𝜒₂,…,𝜒_𝑛)^𝑇,(ii) there exists constant 𝕎≥0 such that lnmax𝜂^𝑚_𝑘=1𝑘,1−𝛿^⋆𝑡_𝑚−𝑡₀≤𝕎∀𝑚∈ℤ₊holds, (3.2)where 𝜂_𝑘 is the largest eigenvalue of 𝑃⁻¹𝐷_𝑘𝑃𝐷_𝑘,𝑘∈ℤ₊.
Then the equilibrium point of the system (2.1) is globally exponentially stable and the approximate exponential convergent rate is (𝜀^⋆−𝛿^⋆)/2.

Proof. We only need to prove that the zero solution of system (2.4) is globally exponentially stable. For any 𝑡₀≥0, let 𝑦(𝑡)=𝑦(𝑡₀,𝜑)(𝑡) be a solution of (2.4) through (𝑡₀,𝜑).
Consider the Lyapunov functional as follows: 𝑉(𝑡)=𝑒^𝜀⋆𝑡𝑦^𝑇(𝑡)(𝑡)+1𝑃𝑦1−𝜌^𝑡_{(𝑡)𝑡−𝜏}𝑒^𝜀⋆𝑠𝐺^𝑇𝑦(𝑠)𝑄₂+𝑇^⋆𝑇𝑇^⋆𝐺𝑦(𝑠)𝑑𝑠, (3.3) then we have 𝜆_min(𝑃)𝑒^𝜀⋆𝑡𝑦(𝑡)²<𝑉(𝑡)≤𝜆_max(𝑃)𝑒^𝜀⋆𝑡𝑦(𝑡)²+𝜆_max𝑄₂+𝑇^⋆𝑇𝑇^⋆𝑁𝑒^𝜀⋆𝑡1−𝑒^{−𝜀⋆𝜏(𝑡)}𝜀^⋆1−𝜌𝑦(𝑡)²_𝜏≤𝜆_max(𝑃)+𝜆_max𝑄₂+𝑇^⋆𝑇𝑇^⋆𝑁1−𝑒^{−𝜀⋆𝜏}𝜀^⋆𝑒1−𝜌^𝜀⋆𝑡𝑦(𝑡)²_𝜏. (3.4) By Lemma 2.5, we get 2𝑦^𝑇(𝑡)𝑃𝐴𝐹𝑦(𝑡)=2𝐹^𝑇𝑦(𝑡)𝐴^𝑇𝑃𝑦(𝑡)≤𝐹^𝑇𝑦(𝑡)𝑄₁𝐹𝑦(𝑡)+𝑦^𝑇(𝑡)𝑃𝐴𝑄₁⁻¹𝐴^𝑇𝑃𝑦(𝑡)≤𝜆_max𝑄₁𝐹^𝑇𝑦(𝑡)𝐹𝑦(𝑡)+𝑦^𝑇(𝑡)𝑃𝐴𝑄₁⁻¹𝐴^𝑇𝑃𝑦(𝑡)≤𝑦^𝑇(𝑡)𝑃𝐴𝑄₁⁻¹𝐴^𝑇𝑃+𝜆_max𝑄₁𝑀𝐸𝑦(𝑡),2𝑦^𝑇(𝑡)𝑃𝐵𝐺𝑦(𝑡−𝜏(𝑡))=2𝐺^𝑇𝑦(𝑡−𝜏(𝑡))𝐵^𝑇(𝑡)𝐺𝑦((𝑡√𝑃𝑦=2𝑡−𝜏))𝑒^{−𝜏𝜀⋆𝑇}𝐵^𝑇𝑃𝑦(𝑡)√𝑒^𝜏𝜀⋆≤𝑒^{−𝜏𝜀⋆}𝐺^𝑇𝑦(𝑡−𝜏(𝑡))𝑄₂𝐺𝑦(𝑡−𝜏(𝑡))+𝑒^𝜏𝜀⋆𝑦^𝑇(𝑡)𝑃𝐵𝑄₂⁻¹𝐵^𝑇𝑃𝑦(𝑡). (3.5)(3.6) On the other hand, since Γ^𝑇Γ=||𝜍||²𝐸 and ||𝜍||≤||𝜒||, then we have 𝑦^𝑇(𝑡)𝑃Γ^𝑇Γ𝑃𝑦(𝑡)≤𝜒²𝑦^𝑇(𝑡)𝑃²𝑦(𝑡), (3.7) where 𝜒=(𝜒₁,𝜒₂,…,𝜒_𝑛)^𝑇.
Thus, we obtain 2𝑦^𝑇(𝑡)𝑃Γ^𝑇𝑇^⋆𝐺𝑦(𝑡−𝜏(𝑡))=2𝐺^𝑇𝑦(𝑡−𝜏(𝑡))𝑇^⋆𝑇(𝑡)𝑇Γ𝑃𝑦=2^⋆𝐺𝑦(𝑡−𝜏(𝑡))√𝑒^{−𝜏𝜀⋆𝑇}Γ𝑃𝑦(𝑡)√𝑒^𝜏𝜀⋆≤𝑒^{−𝜏𝜀⋆}𝐺^𝑇𝑦(𝑡−𝜏(𝑡))𝑇^⋆𝑇𝑇^⋆𝐺𝑦(𝑡−𝜏(𝑡))+𝑒^𝜏𝜀⋆𝑦^𝑇(𝑡)𝑃Γ^𝑇Γ𝑃𝑦(𝑡)≤𝑒^{−𝜏𝜀⋆}𝐺^𝑇𝑦(𝑡−𝜏(𝑡))𝑇^⋆𝑇𝑇^⋆𝐺𝑦(𝑡−𝜏(𝑡))+𝑒^𝜏𝜀⋆𝜒²𝑦^𝑇(𝑡)𝑃²𝑦(𝑡). (3.8) Now we consider the derivation of 𝑉 along the trajectories of system (2.4), for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), 𝑘∈ℤ₊, 𝐷⁺𝑉(𝑡)|_(2.3)=𝑒^𝜀⋆𝑡𝜀^⋆𝑦^𝑇(𝑡)𝑃𝑦(𝑡)+𝑒^𝜀⋆𝑡𝑦^′𝑇(𝑡)𝑃𝑦(𝑡)+𝑦^𝑇(𝑡)𝑃𝑦^′(𝑡+1)𝑒1−𝜌^𝜀⋆𝑡𝐺^𝑇𝑦(𝑡)𝑄₂+𝑇^⋆𝑇𝑇^⋆𝐺𝑦(𝑡)−1−𝜏̇(𝑡)𝑒1−𝜌^{𝜀⋆(𝑡−𝜏(𝑡))}𝐺^𝑇𝑦(𝑡−𝜏(𝑡))𝑄₂+𝑇^⋆𝑇𝑇^⋆𝐺𝑦(𝑡−𝜏(𝑡))≤𝑒^𝜀⋆𝑡𝜀^⋆𝑦^𝑇(𝑡)𝑃𝑦(𝑡)+𝑒^𝜀⋆𝑡𝑦^𝑇(𝑡)(−𝐶𝑃−𝑃𝐶)𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑃𝐴𝐹𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑃𝐵𝐺𝑦(𝑡−𝜏(𝑡))+2𝑦^𝑇(𝑡)𝑃Γ^𝑇𝑇^⋆𝐺𝑦((𝑡+1𝑡−𝜏))𝑒1−𝜌^𝜀⋆𝑡𝐺^𝑇𝑦(𝑡)𝑄₂+𝑇^⋆𝑇𝑇^⋆𝐺𝑦(𝑡)−𝑒^{𝜀⋆(𝑡−𝜏)}𝐺^𝑇𝑦(𝑡−𝜏(𝑡))𝑄₂+𝑇^⋆𝑇𝑇^⋆𝐺𝑦(𝑡−𝜏(𝑡))≤𝑒^𝜀⋆𝑡𝑦^𝑇(𝑡)𝜀^⋆𝑃−𝑃𝐶−𝐶𝑃+𝑃𝐴𝑄₁⁻¹𝐴^𝑇𝑃+𝜆_max𝑄₁+𝑀𝐸𝑁𝜆_max𝑄₂+𝑇^⋆𝑇𝑇^⋆1−𝜌𝐸+𝑒^𝜏𝜀⋆𝑃𝐵𝑄₂⁻¹𝐵^𝑇𝑃+𝑒^𝜏𝜀⋆𝜒²𝑃²𝑦(𝑡)≤0. (3.9) Moreover, we note 𝑉𝑡_𝑘=𝑒^{𝜀⋆𝑡𝑘}𝑦^𝑇𝑡_𝑘𝑃𝑦𝑡_𝑘+11−𝜌^𝑡𝑘_{𝑡𝑘−𝜏(𝑡𝑘)}𝑒^𝜀⋆𝑠𝐺^𝑇𝑦(𝑠)𝑄₂+𝑇^⋆𝑇𝑇^⋆𝐺𝑦(𝑠)𝑑𝑠=𝑒^{𝜀⋆𝑡𝑘}𝑦^𝑇𝑡⁻_𝑘𝐷_𝑘𝑃𝐷_𝑘𝑦𝑡⁻_𝑘+11−𝜌^𝑡−𝑘_𝑡⁻_{𝑘−𝜏(𝑡}⁻_𝑘)𝑒^𝜀⋆𝑠𝐺^𝑇𝑦(𝑠)𝑄₂+𝑇^⋆𝑇𝑇^⋆𝐺𝑦(𝑠)𝑑𝑠≤𝑒^{𝜀⋆𝑡𝑘}𝜂_𝑘𝑦^𝑇𝑡⁻_𝑘𝑃𝑦𝑡⁻_𝑘+11−𝜌^𝑡−𝑘_𝑡⁻_{𝑘−𝜏(𝑡}⁻_𝑘)𝑒^𝜀⋆𝑠𝐺^𝑇𝑦(𝑠)𝑄₂+𝑇^⋆𝑇𝑇^⋆𝐺𝑦(𝑠)𝜂𝑑𝑠≤max_𝑘,1𝑉𝑡⁻_𝑘. (3.10) By simple induction, considering (3.4)–(3.10), we get, for 𝑘≥1, 𝜆_min(𝑃)𝑒^𝜀⋆𝑡𝑦(𝑡)²≤𝑉(𝑡)≤𝑉𝑡_{0𝑡0<𝑡𝑘≤𝑡}max𝜂_𝑘,1. (3.11) On the other hand, from (3.4), we get 𝑉𝑡₀≤𝜆_max(𝑃)+𝜆_max𝑄₂+𝑇^⋆𝑇𝑇^⋆𝑁1−𝑒^{−𝜀⋆𝜏}𝜀^⋆𝑒1−𝜌^𝜀⋆𝑡0𝜑²_𝜏. (3.12) Substituting the above inequality into (3.11), we obtain 𝑦(𝑡)²≤𝜆_max(𝑃)𝜆_min(𝑃)+𝜆_max𝑄₂+𝑇^⋆𝑇𝑇^⋆𝑁1−𝑒^{−𝜀⋆𝜏}𝜀^⋆1−𝜌𝜆_min(𝑃)𝑒^{−𝜀⋆(𝑡−𝑡0)}𝜑²_{𝜏𝑡0<𝑡𝑘≤𝑡}max𝜂_𝑘,1. (3.13) In view of condition (ii), we furthermore have 𝑦(𝑡)≤𝕄𝑒^{−((𝜀⋆−𝛿⋆)/2)(𝑡−𝑡0)}𝜑_𝜏,𝑡≥𝑡₀, (3.14) where 𝕄=𝜆_max(𝑃)𝜆_min(𝑃)+𝜆_max𝑄₂+𝑇^⋆𝑇𝑇^⋆𝑁1−𝑒^{−𝜀⋆𝜏}𝜀^⋆1−𝜌𝜆_min(𝑃)𝑒^𝕎≥1. (3.15) Hence, the zero solution of system (2.4) is globally exponentially stable; that is, the equilibrium point of system (2.1) is globally exponentially stable and the approximate exponential convergent rate is (𝜀^⋆−𝛿^⋆)/2. The proof of Theorem 3.1 is therefore complete.

Remark 3.2. In Theorem 3.1, we find that condition (i) can be replaced by 𝜀^⋆𝑃−𝑃𝐶−𝐶𝑃+𝑃𝐴𝑄₁⁻¹𝐴^𝑇𝑃+𝜆_max𝑄₁𝑀𝐸+𝑁𝜆_max𝑄₂1−𝜌𝐸+𝑁𝜆_max𝑇^⋆𝑇𝑇^⋆𝐸1−𝜌+𝑒^𝜏𝜀⋆𝑃𝐵𝑄₂⁻¹𝐵^𝑇𝑃+𝑒^𝜏𝜀⋆𝜒²𝑃²≤0. (3.16) Leting 𝑃=𝑄₁=𝑄₂=𝐸 in Theorem 3.1, then we have the following.

Corollary 3.3. Assume that there exist constants 𝜀^⋆>0, 𝛿^⋆∈[0,𝜀^⋆) such that(i)  𝜆^⋆_max≤−𝜀^⋆𝑁−𝑀−−1−𝜌𝑁𝜆_max𝑇^⋆𝑇𝑇^⋆1−𝜌−𝑒^𝜏𝜀⋆𝜒², (3.17)where 𝜆^⋆_max is the largest eigenvalue of −2𝐶+𝐴𝐴^𝑇+𝑒^𝜏𝜀⋆𝐵𝐵^𝑇;(ii) there exists constant 𝕎≥0 such that ^𝑚_𝑘=1lnmaxmax_𝑖∈Λ1+𝑑_𝑘^(𝑖)2,1−𝛿^⋆𝑡_𝑚−𝑡₀<𝕎∀𝑚∈ℤ₊holds. (3.18)
The equilibrium point of the system (2.1) is globally exponentially stable and the approximate exponential convergent rate is (𝜀^⋆−𝛿^⋆)/2.

Furthermore, if 𝑑_𝑘^(𝑖)∈[−2,0] in Corollary 3.3, then we have the following result.

Corollary 3.4. The equilibrium point of the system (2.1) is globally exponentially stable, if 𝑑_𝑘^(𝑖)∈[−2,0], and there exists constant 𝜀^⋆>0 such that −2𝐶+𝐴𝐴^𝑇+𝑒^𝜏𝜀⋆𝐵𝐵^𝑇+𝜀^⋆𝑁+𝑀+1+𝜆_max𝑇^⋆𝑇𝑇^⋆1−𝜌+𝑒^𝜏𝜀⋆𝜒²𝐸≤0. (3.19)

Remark 3.5. In fact, Theorem 3.1 implies that if sup_𝑘∈ℤ+∏^𝑘_𝑠=1(1+𝛽_𝑠^(𝑖))²<∞, then one may choose 𝛿^⋆=0. On the other hand, Luo an Cui [13] obtained some results on global asymptotic stability. However, those results cannot ensure the global exponential stability. Let 𝑑_𝑘^(𝑖)=0 (i.e., 𝐷_𝑘=𝐸) in Corollary 3.4, then we can obtain the desirable result as follows.

Corollary 3.6. The equilibrium point of the system (2.1) without impulses is globally exponentially stable, if there exist 𝑛×𝑛 symmetric and positive definite matrices 𝑃, 𝑄₁, 𝑄₂ such that −𝑃𝐶−𝐶𝑃+𝑃𝐴𝑄₁⁻¹𝐴^𝑇𝑃+𝜆_max𝑄₁𝑀𝐸+𝑁𝜆_max𝑄₂+𝑇^⋆𝑇𝑇^⋆𝐸1−𝜌+𝑃𝐵𝑄₂⁻¹𝐵^𝑇𝑃+𝜒²𝑃²<0. (3.20) Furthermore, if 𝑃=𝑄₁=𝑄₂=𝐸 in Corollary 3.6, then it becomes as follows.

Corollary 3.7. The equilibrium point of the system (2.1) without impulses is globally exponentially stable, if the following condition holds: −2𝐶+𝐴𝐴^𝑇+𝐵𝐵^𝑇+𝑁𝜆_max𝑇^⋆𝑇𝑇^⋆+𝑁1−𝜌𝜒1−𝜌+𝑀+²𝐸<0. (3.21)

Remark 3.8. Corollaries 3.6 and 3.7 imply that if the above inequality holds, then there exists enough small 𝜀^⋆>0 such that all conditions in Corollary 3.4 are satisfied. Hence, Corollaries 3.6 and 3.7 supplied a new criteria for global exponential stability of equilibrium point of the system (2.1) without impulses.
Next we can establish a theorem which provide sufficient conditions for uniform stability of system (2.1) by constructing another Lyapunov functional. Here we shall emphasize the effects of impulses.

Theorem 3.9. Assume that there exist 𝑛×𝑛 symmetric and positive definite matrices 𝑃, 𝑄₁, 𝑄₂ such that the following condition −𝑃𝐶−𝐶𝑃+𝑃𝐴𝑄₁⁻¹𝐴^𝑇𝑃+𝜆_max𝑄₁𝑀𝐸+𝑁𝜆_max𝑄₂+𝑇^⋆𝑇𝑇^⋆1−𝜌^𝑘_𝑠=1𝜂_𝑠𝐸+^𝑘_𝑠=1𝜂_𝑠⁻¹𝑃𝐵𝑄₂⁻¹𝐵^𝑇𝑃+^𝑘_𝑠=1𝜂_𝑠⁻¹𝜒²𝑃²≤0∀𝑘∈ℤ₊holds, (3.22) where sup_𝑘∈ℤ+∏^𝑘_𝑠=1𝜂_𝑠<∞, 𝜂_𝑘 is the largest eigenvalue of 𝑃⁻¹𝐷_𝑘𝑃𝐷_𝑘.
Then the equilibrium point of the system (2.1) is uniformly stable.

Proof. We only prove the zero solution of system (2.4) is uniformly stable. For any 𝜀>0, 𝑡₀≥0, 𝜑∈𝑃𝐶_𝛿(𝑡₀), let 𝑦(𝑡)=𝑦(𝑡₀,𝜑)(𝑡) be a solution of (2.4) through (𝑡₀,𝜑), 𝑡₀≥0, then we can prove that ||𝑦(𝑡)||<𝜀, 𝑡≥𝑡₀,
where 𝜀𝛿=𝜆_min(𝑃)√𝜂𝜆_max(𝑃)+𝜆_max𝑄₂+𝑇^⋆𝑇𝑇^⋆𝑁𝜂𝜏/1−𝜌,𝜂=̇sup_𝑘∈ℤ+^𝑘_𝑠=1𝜂_𝑠. (3.23) Consider the following Lyapunov functional 𝑉(𝑡)=𝑦^𝑇(𝑡)(𝑡)+1𝑃𝑦1−𝜌^𝑡_{𝑡−𝜏(𝑡)𝑡𝑠≤𝑡}𝜂_𝑠𝐺^𝑇𝑦(𝑠)𝑄₂+𝑇^⋆𝑇𝑇^⋆𝐺𝑦(𝑠)𝑑𝑠, (3.24) then we have 𝜆_min(𝑃)𝑦(𝑡)²<𝑉(𝑡)≤𝜆_max(𝑃)𝑦(𝑡)²+𝜆_max𝑄₂+𝑇^⋆𝑇𝑇^⋆𝑁𝜂𝜏1−𝜌𝑦(𝑡)²_𝜏≤𝜆_max(𝑃)+𝜆_max𝑄₂+𝑇^⋆𝑇𝑇^⋆𝑁𝜂𝜏1−𝜌𝑦(𝑡)²_𝜏. (3.25) Applying the same argument as Theorem 3.1, we get 2𝑦^𝑇(𝑡)𝑃𝐴𝐹𝑦(𝑡)≤𝑦^𝑇(𝑡)𝑃𝐴𝑄₁⁻¹𝐴^𝑇𝑃+𝜆_max𝑄₁𝑀𝐸𝑦(𝑡),2𝑦^𝑇(𝑡)𝑃𝐵𝐺𝑦(𝑡−𝜏(𝑡))≤^𝑘_𝑠=1𝜂_𝑠𝐺^𝑇𝑦(𝑡−𝜏(𝑡))𝑄₂𝐺𝑦((𝑡+𝑡−𝜏))_𝑘𝑠=1𝜂_𝑠⁻¹𝑦^𝑇(𝑡)𝑃𝐵𝑄₂⁻¹𝐵^𝑇𝑃𝑦(𝑡),2𝑦^𝑇(𝑡)𝑃Γ^𝑇𝑇^⋆𝐺𝑦(𝑡−𝜏(𝑡))≤^𝑘_𝑠=1𝜂_𝑠𝐺^𝑇𝑦(𝑡−𝜏(𝑡))𝑇^⋆𝑇𝑇^⋆𝐺𝑦((𝑡+𝑡−𝜏))_𝑘𝑠=1𝜂_𝑠⁻¹𝜒²𝑦^𝑇(𝑡)𝑃²𝑦(𝑡). (3.26) By simple calculation, we can obtain, for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1),𝑘∈ℤ₊, 𝐷⁺𝑉(𝑡)|_(2.3)=𝑦^𝑇(𝑡)(−𝐶𝑃−𝑃𝐶)𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑃𝐴𝐹𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑃𝐵𝐺𝑦(𝑡−𝜏(𝑡))+2𝑦^𝑇(𝑡)𝑃Γ^𝑇𝑇^⋆𝐺𝑦((𝑡+1𝑡−𝜏))1−𝜌^𝑘_𝑠=1𝜂_𝑠𝐺^𝑇𝑦(𝑡)𝑄₂+𝑇^⋆𝑇𝑇^⋆𝐺𝑦(𝑡)−1−𝜏̇(𝑡)1−𝜌^𝑘_𝑠=1𝜂_𝑠𝐺^𝑇𝑦(𝑡−𝜏(𝑡))𝑄₂+𝑇^⋆𝑇𝑇^⋆𝐺𝑦(𝑡−𝜏(𝑡))≤𝑦^𝑇(𝑡)−𝑃𝐶−𝐶𝑃+𝑃𝐴𝑄₁⁻¹𝐴^𝑇𝑃+𝜆_max𝑄₁+𝑀𝐸𝑁𝜆_max𝑄₂+𝑇^⋆𝑇𝑇^⋆1−𝜌^𝑘_𝑠=1𝜂_𝑠𝐸+^𝑘_𝑠=1𝜂_𝑠⁻¹𝑃𝐵𝑄₂⁻¹𝐵^𝑇𝑃+^𝑘_𝑠=1𝜂_𝑠⁻¹𝜒²𝑃²𝑦(𝑡).≤0. (3.27) Moreover, we know 𝑉𝑡_𝑘=𝑦^𝑇𝑡_𝑘𝑃𝑦𝑡_𝑘+11−𝜌^𝑡𝑘_{𝑡𝑘−𝜏(𝑡𝑘)𝑡𝑠≤𝑡𝑘}𝜂_𝑠Γ^𝑇𝑦(𝑠)𝑄₂+𝑇^⋆𝑇𝑇^⋆Γ𝑦(𝑠)𝑑𝑠=𝑦^𝑇𝑡⁻_𝑘𝐷_𝑘𝑃𝐷_𝑘𝑦𝑡⁻_𝑘+11−𝜌^𝑡−𝑘_𝑡⁻_{𝑘−𝜏(𝑡𝑘)𝑡𝑠≤𝑡𝑘}𝜂_𝑠Γ^𝑇𝑦(𝑠)𝑄₂+𝑇^⋆𝑇𝑇^⋆Γ𝑦(𝑠)𝑑𝑠≤𝜂_𝑘𝑦^𝑇𝑡⁻_𝑘𝑃𝑦𝑡⁻_𝑘+1𝜂1−𝜌^{𝑘𝑡−𝑘}_𝑡⁻_{𝑘−𝜏(𝑡𝑘)𝑡𝑠≤𝑡𝑘−1}𝜂_𝑠Γ^𝑇𝑦(𝑠)𝑄₂+𝑇^⋆𝑇𝑇^⋆Γ𝑦(𝑠)𝑑𝑠=𝜂_𝑘𝑉𝑡⁻_𝑘. (3.28) By simple induction, from (3.27) and (3.28) we may prove that, for 𝑘≥1, 𝜆_min(𝑃)𝑦(𝑡)²≤𝑉(𝑡)≤𝑉𝑡_{0𝑡0<𝑡𝑘≤𝑡}𝜂_𝑘. (3.29) Employing the fact (3.25), we obtain 𝜆_min(𝑃)𝑦(𝑡)²≤𝜆_max(𝑃)+𝜆_max𝑄₂+𝑇^⋆𝑇𝑇^⋆𝑁𝜂𝜏𝜑1−𝜌^2𝜏𝜂,𝑡≥𝑡₀, (3.30) which implies that 𝑦(𝑡)<𝜀,𝑡≥𝑡₀. (3.31) Therefore, the zero solution of system (2.4) is uniformly stable, that is, the equilibrium point of system (2.1) is uniformly stable. The proof of Theorem 3.9 is complete.

Corollary 3.10. The equilibrium point of the system (2.1) is uniformly stable, if there exist 𝑛×𝑛 symmetric and positive definite matrices 𝑃, 𝑄₁, 𝑄₂ such that the following condition holds: Ξ_𝑘≤−𝜆_max𝑄₁𝑀−𝑁𝜆_max𝑄₂+𝑇^⋆𝑇𝑇^⋆1−𝜌^𝑘_𝑠=1𝜂_𝑠, (3.32) where Ξ_𝑘 is the largest eigenvalue of −𝑃𝐶−𝐶𝑃+𝑃𝐴𝑄₁⁻¹𝐴^𝑇𝑃+(∏^𝑘_𝑠=1𝜂_𝑠)⁻¹[𝑃𝐵𝑄₂⁻¹𝐵^𝑇𝑃+||𝜒||²𝑃²].

If 𝑃=𝑄₁=𝑄₂=𝐸 in Theorem 3.9, then we have the following.

Corollary 3.11. The equilibrium point of the system (2.1) is uniformly stable, if the following condition −2𝐶+𝐴𝐴^𝑇+𝑀+𝑁𝜆_max𝐸+𝑇^⋆𝑇𝑇^⋆1−𝜌^𝑘∏_𝑠=1max_𝑖∈Λ1+𝑑_𝑠^(𝑖)2𝐸+^𝑘_𝑠=1max_𝑖∈Λ1+𝑑_𝑠^(𝑖)2−1𝐵𝐵^𝑇+𝜒²𝐸≤0∀𝑘∈ℤ₊holds, (3.33) where sup_𝑘∈ℤ+∏^𝑘_𝑠=1max_𝑖∈Λ(1+𝑑_𝑠^(𝑖))²<∞.

4. Examples

In this section we give two examples to demonstrate our results.

Example 4.1. Consider the following high-order delayed Hopfield-type neural network with impulses 𝑥^′_𝑖(𝑡)=−𝑐_𝑖𝑥_𝑖(𝑡)+³_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+³_𝑗=1𝑏_𝑖𝑗𝑔_𝑗𝑥_𝑗((𝑡+𝑡−𝜏))_{33𝑗=1𝑙=1}𝑇_𝑖𝑗𝑙𝑔_𝑙𝑥_𝑙(𝑡−𝜏(𝑡))𝑔_𝑗𝑥_𝑗(𝑡−𝜏(𝑡)),𝑡≠𝑡_𝑘,𝑡≥𝑡₀,Δ𝑥_𝑖|_{𝑡=𝑡𝑘}=𝑥_𝑖𝑡_𝑘−𝑥_𝑖𝑡⁻_𝑘=𝛽_𝑘^(𝑖)𝑥_𝑖𝑡⁻_𝑘,𝑖=1,2,3,𝑘∈ℤ₊, (4.1) where 𝛽_𝑘^(𝑖)=1+(𝑖/𝑘²)−1, 𝜏(𝑡)=sin𝑡/2, 𝑓₁(𝑥₁)=tanh(0.5𝑥₁), 𝑓₂(𝑥₂)=tanh(0.48𝑥₂), 𝑓₃(𝑥₃)=tanh(0.6𝑥₃), 𝑔₁(𝑥₁)=tanh(0.3𝑥₁), 𝑔₂(𝑥₂)=tanh(0.8𝑥₂), 𝑔₃(𝑥₃)=tanh(0.73𝑥₃), 𝐶=diag𝑐₁,𝑐₂,𝑐₃^𝑇=𝑎3.20002.50002.0,𝐴=_𝑖𝑗3×3=,𝑏0.580.120.23−0.080.36−0.05−0.040.04−0.37𝐵=_𝑖𝑗3×3=0.0600.040.19−0.17−0.02−0.030.130.44,𝑇₁=𝑇_{1𝑗𝑙3×3}=,𝑇0.03−0.20−0.05−0.06−0.140.230.270.03−0.20₂=𝑇_{2𝑗𝑙3×3}=0.01−0.050.08−0.06−0.03−0.090.15−0.040.11,𝑇₃=𝑇_{3𝑗𝑙3×3}=.−0.02−0.12−0.050.240.040.07−0.020.080.01 (4.2) In this case, we easily observe that 𝜏=𝜌=0.5, 𝑀=0.36, 𝑁=0.64, ||𝜒||²=0.64.
For Theorem 3.1, choosing 𝑃=𝑄₁=𝑄₂=𝐸, then from ^∞_𝑘=11+𝛽_𝑘^(𝑖)2=^∞_𝑘=1𝑖1+𝑘²<∞,𝑖=1,2,3, (4.3) we may choose 𝜀^⋆=0.0976, 𝛿^⋆=0,𝕎=∏^∞_𝑘=1(1+3/𝑘²)<∞.
On the other hand, we can compute −2𝐶+𝐴𝐴^𝑇+𝑒^𝜏𝜀⋆𝐵𝐵^𝑇=−5.9908−0.0036−0.0869−0.0036−4.7928−0.0023−0.0869−0.0023−3.6379=Θ (4.4) which implies that 𝜆_max(Θ)=−3.6347.
Also, we note that 𝑇^⋆𝑇𝑇^⋆=𝑇₁+𝑇^𝑇₁,𝑇₂+𝑇^𝑇₂,𝑇₃+𝑇^𝑇₃^𝑇=0.20590.0832−0.05350.08320.2895−0.2735−0.0535−0.27350.4220 (4.5) implies that −𝜀^⋆𝑁−𝑀−−1−𝜌𝑁𝜆_max𝑇^⋆𝑇𝑇^⋆1−𝜌−𝑒^𝜏𝜀⋆𝜒²≈−3.2501>−3.6347. (4.6) By Corollary 3.3, the equilibrium point of (4.1) (0,0,0)^𝑇 is global exponential stable with the approximate convergence rate 0.0488.
However, the criteria in [12] are invalid here.

Example 4.2. Consider the high-order delayed Hopfield-type neural network with impulses [13] 𝑥^′_𝑖(𝑡)=−𝑐_𝑖𝑥_𝑖(𝑡)+²_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+²_𝑗=1𝑏_𝑖𝑗𝑔_𝑗𝑥_𝑗((𝑡+𝑡−𝜏))_{22𝑗=1𝑙=1}𝑇_𝑖𝑗𝑙𝑔_𝑙𝑥_𝑙(𝑡−𝜏(𝑡))𝑔_𝑗𝑥_𝑗(𝑡−𝜏(𝑡))+𝐼_𝑖,𝑡≠𝑡_𝑘,𝑡≥𝑡₀, (4.7) and with impulses Δ𝑥_𝑖|_{𝑡=𝑡𝑘}=𝑥_𝑖𝑡_𝑘−𝑥_𝑖𝑡⁻_𝑘=𝛽_𝑘^(𝑖)𝑥_𝑖𝑡⁻_𝑘−𝑥^∗,𝑖=1,2,𝑘∈ℤ₊, (4.8) where 𝑡_𝑘=𝑘, 𝑡₀=0, 𝑘∈ℤ₊, 𝑓₁(𝑥₁)=𝑔₁(𝑥₁)=tanh(0.53𝑥₁), 𝑓₂(𝑥₂)=𝑔₂(𝑥₂)=tanh(0.67𝑥₂), 𝜌=0.6, 𝐽₁=1.5, 𝐽₂=2, 𝐶=diag𝑐₁,𝑐₂^𝑇=𝑎1.9001.89,𝐴=_𝑖𝑗2×2=,𝑏0.050.140.200.31𝐵=_𝑖𝑗2×2=0.090.250.210.45,𝑇₁=𝑇_{1𝑗𝑙2×2}=,𝑇0.050.14−0.060.05₂=𝑇_{2𝑗𝑙2×2}=,𝛽0.290.100.23−0.14_𝑘(𝑖)=𝑒^0.0625−1,𝑘∈ℤ₊. (4.9) It is obvious that 𝑀=𝑁=𝐿²=0.4489, ||𝜒||²=0.4489. Here we consider 𝜏=1. Choose 𝑃=𝑄₁=𝑄₂=𝐸, 𝜀^⋆=0.25, 𝛿^⋆=0.128.
Note that ^𝑚_𝑘=1lnmaxmax_𝑖∈Λ1+𝑑_𝑘^(𝑖)2,1−𝛿^⋆𝑡_𝑚−𝑡₀=0.125𝑚−0.128𝑚=−0.003𝑚<0∀𝑚∈ℤ₊holds. (4.10) On the other hand, we can compute −2𝐶+𝐴𝐴^𝑇+𝑒^𝜏𝜀⋆𝐵𝐵^𝑇=−3.70730.18480.1848−3.3973=Δ (4.11) which implies that 𝜆_max(Δ)=−3.3111.
One can check that −𝜀^⋆𝑁−𝑀−−1−𝜌𝑁𝜆_max𝑇^⋆𝑇𝑇^⋆1−𝜌−𝑒^𝜏𝜀⋆𝜒²≈−2.9649>−3.3111. (4.12) By Corollary 3.3, the equilibrium point of (4.1)–(4.7) 𝑥^∗ is global exponential stable with the approximate convergence rate 0.061.
In fact, for above-given impulsive condition, we only need time-delay 𝜏 which satisfies the following condition: 𝜆_max−2𝐶+𝐴𝐴^𝑇+𝑒^0.125𝜏𝐵𝐵^𝑇<−0.4489𝑒^0.125𝜏−2.2635. (4.13)

Remark 4.3. In [13], the author obtained that the equilibrium point of (4.7) without impulses is globally asymptotically stable. From the example, we obtain that the equilibrium point of (4.7) without impulses is global exponential stability. In fact, if 𝛽_𝑘^(𝑖)=0 in (4.7), then we can choose 𝛿^⋆=0, which implies that, for any given 𝜏>0, there exists corresponding 𝜀^⋆>0 such that all conditions in Corollary 3.6 are satisfied.

5. Conclusions

In this paper, a class of high-order delayed HNN with impulses is considered. We obtain some new criteria ensuring global exponential stability and uniform stability of the equilibrium point for such system by using Lyapunov functional method, the quality of negative definite matrix, and the linear matrix inequality. Our results show delays and impulsive effects on the stability of HNN. Two examples are given to illustrate the feasibility of the results.

Acknowledgments

This work was jointly supported by the Project of Shandong Province Higher Educational Science and Technology Program (no. J12LN23), Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (no. BS2012DX038), National Science Foundation for Postdoctoral Scientists of China (no. 2012M511538), and National Natural Science Foundation of China (no. 61201441).